Philosophy of Mathematics, Misc

The aim of this paper is to present a constructive solution to Frege's puzzle (largely limited to the mathematical context) based on type theory. Two ways in which an equality statement may be said to have cognitive significance are distinguished. One concerns the mode of presentation of the equality, the other its mode of proof. Frege's distinction between sense and reference, which emphasizes the former aspect, cannot adequately explain the cognitive significance of equality statements unless a clear identity criterion for (...) senses is provided. It is argued that providing a solution based on proofs is more satisfactory from the standpoint of constructive semantics. (shrink)

Último sermão da Igreja do naturalismo fundamentalista pelo pastor Hofstadter. Como o seu muito mais famoso (ou infame por seus erros filosóficos implacáveis) Godel, Escher, Bach, ele tem uma plausibilidade superficial, mas se se compreende que este é um cientificismo desenfreado que mistura questões científicas reais com os filosóficos (ou seja, o somente as edições reais são que jogos da língua nós devemos jogar) então quase todo seu interesse desaparece. Eu forneci um quadro para análise baseada na psicologia evolutiva e (...) no trabalho de Wittgenstein (desde que atualizado em meus escritos mais recentes). Aqueles que desejam um quadro até à data detalhado para o comportamento humano da opinião moderna dos dois sistemas consultar meu livros Falando Macacos 3ª Ed (2019), A Estrutura Lógica da Filosofia, Psicologia, Mente e Linguagem em Ludwig Wittgenstein e John Searle 2a Ed (2019), Suicídio Pela Democracia,4aEd (2019), Entendendo as Conexões entre Ciência, Filosofia, Psicologia, Religião, Política e Economia Artigos e Análises 2006-2019 (2019), Ilusões Utópicas Suicidas no século 21 6a Ed (2020), A Estrutura Lógica do Comportamento Humano (2019), e A Estrutura Lógica da Consciência (2019) y outras. "Pode ser justamente perguntado qual a importância que a prova de Gödel tem para o nosso trabalho. Para um pedaço de matemática não pode resolver problemas do tipo que nos incomoda. --A resposta é que a situação, em que tal prova nos traz, é do nosso interesse. "O que devemos dizer agora?" -Esse é o nosso tema. No entanto, estranho soa, minha tarefa, tanto quanto diz respeito à prova de Gödel parece meramente consistir em deixar claro que tal proposição como: "suponha que isso poderia ser provado" significa em matemática. " Wittgenstein "observações sobre as fundações da matemática" p337 (1956) (escrito em 1937). "Meus teoremas só mostram que a mecanização da matemática, ou seja, a eliminação da mente e de entidades abstratas, é impossível, se alguém quiser ter uma base satisfatória e um sistema de matemática. Eu não tenho provado que existem questões matemáticas que são indecidíveis para a mente humana, mas só que não há nenhuma máquina (ou formalismo cego) que pode decidir todas as questões número-teórico, (mesmo de um tipo muito especial).... Não é a própria estrutura dos sistemas dedutivos que está a ser ameaçado com um demolir, mas apenas uma certa interpretação do mesmo, ou seja, a sua interpretação como um formalismo cego. " Gödel "obras coletadas" Vol. 5, p 176-177. (2003) "Toda inferência tem lugar a priori. Os acontecimentos do futuro não podem ser inferidos a partir do presente. A superstição é a crença no nexo causal. A liberdade da vontade consiste no fato de que as ações futuras não podem ser conhecidas agora. Só podíamos conhecê-los se a causalidade fosse uma necessidade interior, como a da dedução lógica. --A conexão do conhecimento e do que é sabido é aquela da necessidade lógica. ("A sabe que p é o caso" é sem sentido se p é um tautologia.) Se do fato que uma proposição é óbvia a nós, não segue que é verdadeiro, a seguir o obviedade não é nenhuma justificação para a crença em sua verdade. " TLP 5,133--5,1363 . (shrink)

Eu dou uma revisão detalhada de "os limites exteriores da razão" por Noson Yanofsky de uma perspectiva unificada de Wittgenstein e psicologia evolutiva. Eu indico que a dificuldade com tais questões como paradoxo na linguagem e matemática, incompletude, undecidabilidade, computabilidade, o cérebro eo universo como computadores, etc., todos surgem a partir da falta de olhar atentamente para o nosso uso da linguagem no apropriado contexto e, consequentemente, a falta de separar questões de fato científico a partir de questões de como (...) a linguagem funciona. Discuto os pontos de vista de Wittgenstein sobre incompletude, paraconsistência e indecidabilidade e o trabalho de Wolpert sobre os limites para a computação. Para resumir: o universo de acordo com o Brooklyn---boa ciência, não tão boa filosofia. Aqueles que desejam um quadro até à data detalhado para o comportamento humano da opinião moderna dos dois sistemas consultar meu livros Falando Macacos 3ª Ed (2019), A Estrutura Lógica da Filosofia, Psicologia, Mente e Linguagem em Ludwig Wittgenstein e John Searle 2a Ed (2019), Suicídio Pela Democracia,4aEd(2019), Entendendo as Conexões entre Ciência, Filosofia, Psicologia, Religião, Política e Economia Artigos e Análises 2006-2019 (2019), Ilusões Utópicas Suicidas no 21St século 5a Ed (2019), A Estrutura Lógica do Comportamento Humano (2019), e A Estrutura Lógica da Consciência (2019) y outras. (shrink)

Em "Godel's Way", três cientistas eminentes discutem questões como a undecidability, incompletude, aleatoriedade, computabilidade e paraconsistência. Eu abordar estas questões do ponto de vista Wittgensteinian que existem duas questões básicas que têm soluções completamente diferentes. Há as questões científicas ou empíricas, que são fatos sobre o mundo que precisam ser investigados observacionalmente e questões filosóficas sobre como a linguagem pode ser usada inteligìvelmente (que incluem certas questões em matemática e lógica), que precisam ser decidido por olhar uma como nós realmente (...) usar palavras em contextos específicos. Quando nós começ claros sobre que jogo da língua nós estamos jogando, estes tópicos são vistos para ser perguntas científicas e matemáticas ordinárias como qualquer outro. As idéias de Wittgenstein raramente foram igualadas e nunca ultrapassaram e são tão pertinentes hoje como eram 80 anos atrás, quando ele ditou os livros azul e marrom. Apesar de suas falhas-realmente uma série de notas em vez de um livro acabado-esta é uma fonte única do trabalho destes três estudiosos famosos que têm trabalhado nas bordas sangrantes da física, matemática e filosofia por mais de meio século. Da costa e Doria são citados por Wolpert (veja abaixo ou meus artigos sobre Wolpert e minha revisão de Yanofsky ' s "os limites exteriores da razão") desde que escreveu sobre a computação universal, e entre suas muitas realizações, da costa é um pioneiro em a paraconsistência. Aqueles que desejam um quadro até à data detalhado para o comportamento humano da opinião moderna dos dois sistemas consultar meu livros Falando Macacos 3ª Ed (2019), A Estrutura Lógica da Filosofia, Psicologia, Mente e Linguagem em Ludwig Wittgenstein e John Searle 2a Ed (2019), Suicídio Pela Democracia,4aEd(2019), Entendendo as Conexões entre Ciência, Filosofia, Psicologia, Religião, Política e Economia Artigos e Análises 2006-2019 (2019), Ilusões Utópicas Suicidas no 21St século 5a Ed (2019), A Estrutura Lógica do Comportamento Humano (2019), e A Estrutura Lógica da Consciência (2019) y outras. (shrink)

In aceasta lucrare, autorul si-a propus sa readuca in discutie problema aplicabilitatii matematicii - atat de importanta pentru Kant, dar care a ajuns sa fie in mare parte neglijata in filosofia contemporana a matematicii - incercand sa argumenteze, plecand de la ea, pentru o viziune empirista in filosofia matematicii. Strategia pe care autorul isi propune sa o urmeze, in linii mari, consta intr-o incercare de a atrage atentia asupra unei probleme care apare atunci cand avem in vedere utilitatea matematicii in (...) fizica (De ce este posibil ca matematica sa-l serveasca atat de bine pe fizician? Daca matematica are un subiect propriu de studiu si metode proprii de studiere a acestuia si evolueaza conform propriei sale dinamici interne, cum de poate juca un rol atat de important, dupa unii indispensabil, pentru stiinta?) - problema care este diferita de provocarea lui Wigner si a lui Steiner si de problema coordonarii care apare la pozitivistii logici - urmata de un argument pentru ideea ca singura cale de a raspunde la aceasta problema ar fi aceea de a adopta o doctrina empirista in filosofia matematicii. (shrink)

Doy una revisión detallada de ' los límites externos de la razón ' por Noson Yanofsky desde una perspectiva unificada de Wittgenstein y la psicología evolutiva. Yo indiqué que la dificultad con cuestiones como la paradoja en el lenguaje y las matemáticas, la incompletitud, la indeterminación, la computabilidad, el cerebro y el universo como ordenadores, etc., surgen de la falta de mirada cuidadosa a nuestro uso del lenguaje en el adecuado contexto y, por tanto, el Error al separar los problemas (...) de hecho científico de las cuestiones de cómo funciona el lenguaje. Discuto las opiniones de Wittgenstein sobre la incompletitud, la paracoherencia y la indecisión y el trabajo de Wolpert en los límites de la computación. Resumiendo: el universo según Brooklyn---buena ciencia, no tan buena filosofía. Aquellos que deseen un marco completo hasta la fecha para el comportamiento humano de la moderna dos sistemas punto de vista puede consultar mi libros Talking Monkeys 3ª ed (2019), Estructura Logica de Filosofia, Psicología, Mente y Lenguaje en Ludwig Wittgenstein y John Searle 2a ed (2019), Suicidio pela Democracia 4ª ed (2019), La Estructura Logica del Comportamiento Humano (2019), The Logical Structure de la Conciencia (2019, Entender las Conexiones entre Ciencia, Filosofía, Psicología, Religión, Política y Economía y Delirios Utópicos Suicidas en el siglo 21 5ª ed (2019), Observaciones sobre Imposibilidad, Incompletitud, Paraconsistencia, Indecidibilidad, Aleatoriedad, Computabilidad, Paradoja e Incertidumbre en Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych Berto, Floyd, Moyal-Sharrock y Yanofsky. (shrink)

En ' Godel’s Way ', tres eminentes científicos discuten temas como la indecisión, la incompleta, la aleatoriedad, la computabilidad y la paraconsistencia. Me acerco a estas cuestiones desde el punto de vista de Wittgensteinian de que hay dos cuestiones básicas que tienen soluciones completamente diferentes. Existen las cuestiones científicas o empíricas, que son hechos sobre el mundo que necesitan ser investigados observacionalmente y cuestiones filosóficas en cuanto a cómo el lenguaje se puede utilizar inteligiblemente (que incluyen ciertas preguntas en matemáticas (...) y lógica), que necesitan decidirse por unt cómo realmente usar palabras en contextos concretos. Cuando tenemos claro sobre qué juego de idiomas estamos jugando, estos temas son vistos como preguntas científicas y matemáticas ordinarias como cualquier otra. Las percepciones de Wittgenstein rara vez se han igualado y nunca superado y son tan pertinentes hoy como lo fueron hace 80 años cuando dictó los libros azul y marrón. A pesar de sus fallas — realmente una serie de notas en lugar de un libro terminado —, esta es una fuente única de la obra de estos tres eruditos famosos que han estado trabajando en los bordes sangrantes de la física, las matemáticas y la filosofía durante más de medio siglo. Da Costa y Doria son citados por Wolpert (ver abajo o mis artículos sobre Wolpert y mi reseña de ' los límites de la razón ' de Yanofsky) desde que escribieron en el cómputo universal, y entre sus muchos logros, da Costa es pionera en paraconsistencia. Aquellos que deseen un marco completo hasta la fecha para el comportamiento humano de la moderna dos sistemas punto de vista puede consultar mi libros Talking Monkeys 3ª ed (2019), Estructura Logica de Filosofia, Psicología, Mente y Lenguaje en Ludwig Wittgenstein y John Searle 2a ed (2019), Suicidio pela Democracia 4ª ed (2019), La Estructura Logica del Comportamiento Humano (2019), The Logical Structure de la Conciencia (2019, Entender las Conexiones entre Ciencia, Filosofía, Psicología, Religión, Política y Economía y Delirios Utópicos Suicidas en el siglo 21 5ª ed (2019), Observaciones sobre Imposibilidad, Incompletitud, Paraconsistencia, Indecidibilidad, Aleatoriedad, Computabilidad, Paradoja e Incertidumbre en Chaitin, Wittgenstein,. (shrink)

It is commonly thought that Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were mostly resolved by Wittgenstein over 80years ago. -/- “What we are ‘tempted to say’ in such a case is, of course, not philosophy, but it is its raw material. Thus, for example, what a mathematician is (...) inclined to say about the objectivity and reality of mathematical facts, is not a philosophy of mathematics, but something for philosophical treatment.” Wittgenstein PI 234 -/- "Philosophers constantly see the method of science before their eyes and are irresistibly tempted to ask and answer questions in the way science does. This tendency is the real source of metaphysics and leads the philosopher into complete darkness." Wittgenstein -/- I provide a brief summary of some of the major findings of two of the most eminent students of behavior of modern times, Ludwig Wittgenstein and John Searle, on the logical structure of intentionality (mind, language, behavior), taking as my starting point Wittgenstein’s fundamental discovery –that all truly ‘philosophical’ problems are the same—confusions about how to use language in a particular context, and so all solutions are the same—looking at how language can be used in the context at issue so that its truth conditions (Conditions of Satisfaction or COS) are clear. The basic problem is that one can say anything, but one cannot mean (state clear COS for) any arbitrary utterance and meaning is only possible in a very specific context. -/- I dissect some writings of a few of the major commentators on these issues from a Wittgensteinian viewpoint in the framework of the modern perspective of the two systems of thought (popularized as ‘thinking fast, thinking slow’), employing a new table of intentionality and new dual systems nomenclature. I show that this is a powerful heuristic for describing the true nature of these putative scientific, physical or mathematical issues which are really best approached as standard philosophical problems of how language is to be used (language games in Wittgenstein’s terminology). -/- It is my contention that the table of intentionality (rationality, mind, thought, language, personality etc.) that features prominently here describes more or less accurately, or at least serves as an heuristic for, how we think and behave, and so it encompasses not merely philosophy and psychology, but everything else (history, literature, mathematics, politics etc.). Note especially that intentionality and rationality as I (along with Searle, Wittgenstein and others) view it, includes both conscious deliberative linguistic System 2 and unconscious automated prelinguistic System 1 actions or reflexes. (shrink)

Último sermón de la iglesia del naturalismo fundamentalista por el pastor Hofstadter. Al igual que su mucho más famoso (o infame por sus incesantemente errores filosóficos) trabajo Godel, Escher, Bach, tiene una plausibilidad superficial, pero si se entiende que se trata de un científico rampante que mezcla problemas científicos reales con los filosóficos (es decir, el sólo los problemas reales son los juegos de idiomas que debemos jugar) entonces casi todo su interés desaparece. Proporciono un marco para el análisis basado (...) en la psicología evolutiva y el trabajo de Wittgenstein (ya actualizado en mis escritos más recientes). Aquellos que deseen un marco completo hasta la fecha para el comportamiento humano de la moderna dos sistemas punto de vista puede consultar mi libros Talking Monkeys 3ª ed (2019), Estructura Logica de Filosofia, Psicología, Mente y Lenguaje en Ludwig Wittgenstein y John Searle 2a ed (2019), Suicidio pela Democracia 4ª ed (2019), La Estructura Logica del Comportamiento Humano (2019), The Logical Structure de la Conciencia (2019, Entender las Conexiones entre Ciencia, Filosofía, Psicología, Religión, Política y Economía y Delirios Utópicos Suicidas en el siglo 21 5ª ed (2019), Observaciones sobre Imposibilidad, Incompletitud, Paraconsistencia, Indecidibilidad, Aleatoriedad, Computabilidad, Paradoja e Incertidumbre en Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych Berto, Floyd, Moyal-Sharrock y Yanofsky y otras. (shrink)

The content of the manuscript represents a bold idea system, it is beyond the boundaries of all existing knowledge but the method of reasoning and logic is also very strict and scientific. The purpose of the manuscript is to unify the natural categories (natural philosophy, natural geometry, quantum mechanics, astronomy,…), and to open a new direction for most other sciences. Abstract of the manuscript: About Philosophy: • Proved the existence of time and non-dilation. • Proved that matter is always motionless (...) in space. • Conclusion that space is energy. • … About Mathematics: • Solved the squaring the circle problem(a millennium problem). • Using the equation to calculate speed of light is 471.000.000 m/s • … About Physics : • Explained the nature of gravity. • Explained dark energy, spin value of Nucleon. • Explained matter and antimatter. • …. (shrink)

New Translation of Henri Poincaré's Science and Hypothesis, including new material and editorial commentary. New Introduction by David J. Stump.

This volume contains thirteen papers that were presented at the 2017 Annual Meeting of the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques, which was held at Ryerson University in Toronto. It showcases rigorously reviewed modern scholarship on an interesting variety of topics in the history and philosophy of mathematics from Ancient Greece to the twentieth century. -/- A series of chapters all set in the eighteenth century consider topics such as John Marsh’s (...) techniques for the computation of decimal fractions, Euler’s efforts to compute the surface area of scalene cones, a little-known work by John Playfair on the practical aspects of mathematics, and Monge’s use of descriptive geometry. -/- After a brief stop in the nineteenth century to consider the culture of research mathematics in 1860s Prussia, the book moves into the twentieth century with an examination of the historical context within which the Axiom of Choice was developed and a paper discussing Anatoly Vlasov’s adaptation of the Boltzmann equation to ionized gases. The remaining chapters deal with the philosophy of twentieth-century mathematics through topics such as an historically informed discussion of finitism and its limits; a reexamination of Mary Leng’s defenses of mathematical fictionalism through an alternative, anti-realist approach to mathematics; and a look at the reasons that mathematicians select specific problems to pursue. -/- Written by leading scholars in the field, these papers are accessible to not only mathematicians and students of the history and philosophy of mathematics, but also anyone with a general interest in mathematics. (shrink)

This paper introduces a novel object that has less structure than, and is ontologically prior to the natural numbers. As such it is a candidate model of the foundation that lies beneath the natural numbers. The implications for the construction of mathematical objects built upon that foundation are discussed.

In a series of papers Ladyman and Presnell raise an interesting challenge of providing a pre-mathematical justification for homotopy type theory. In response, they propose what they claim to be an informal semantics for homotopy type theory where types and terms are regarded as mathematical concepts. The aim of this paper is to raise some issues which need to be resolved for the successful development of their types-as-concepts interpretation.

K. Marx’s 200th jubilee coincides with the celebration of the 85 years from the first publication of his “Mathematical Manuscripts” in 1933. Its editor, Sofia Alexandrovna Yanovskaya (1896–1966), was a renowned Soviet mathematician, whose significant studies on the foundations of mathematics and mathematical logic, as well as on the history and philosophy of mathematics are unduly neglected nowadays. Yanovskaya, as a militant Marxist, was actively engaged in the ideological confrontation with idealism and its influence on modern mathematics and their interpretation. (...) Concomitantly, she was one of the pioneers of mathematical logic in the Soviet Union, in an era of fierce disputes on its compatibility with Marxist philosophy. Yanovskaya managed to embrace in an originally Marxist spirit the contemporary level of logico-philosophical research of her time. Due to her highly esteemed status within Soviet academia, she became one of the most significant pillars for the culmination of modern mathematics in the Soviet Union. In this paper, I attempt to trace the influence of the complex socio-cultural context of the first decades of the Soviet Union on Yanovskaya’s work. Among the several issues I discuss, her encounter with L. Wittgenstein is striking. (shrink)

The latest book of Reviel Netz presents a highly erudite analysis of the style of Hellenistic mathematics. Besides the Introduction and Conclusion the book is composed of four chapters. Before turning to more general remarks I would like first to outline the contents of the book.The Introduction starts with the presentation of Archimedes’ Spiral lines. In a condensed form, the author outlines his approach and calls attention to the stylistic peculiarities of that work. Most of the themes discussed in more (...) detail in the following chapters, such as the analysis of the narrative structure of the treatise, its working with surprise, the exposition of the rhetorical devices, the discussion of technical tools, and the breaking of genre-boundaries emerge here side by side in a single treatise.The first chapter, called The carnival of calculation, starts with another work of Archimedes, the Stomachion, for which Netz offers a new interpretation as a treatise in geometrical combinatorics. Then he turns to Archimedes’ On the measurement of the circle, where, in a detailed analysis of the calculations made by Archimedes, Netz presents a pattern that he identifies in several other works of Hellenistic mathematics, a pattern he calls ‘asymptotic calculation’. It consists in a combination of strict geometric arguments with calculations of proportions, some precise, some only approximate, which together with rather arbitrary simplifications lead to a result that is achieved after an abrupt and unexpected halt. Netz shows that, according to the surviving testimonies, a similar pattern can be found in many Hellenistic authors, such as Aristarchus, Hipparchus, and Apollonius of Perga. "[I]n many Hellenistic mathematical works we see calculations that give rise to values that are big and unwieldy, or else are only asymptotically found. Such calculations end up not with the satisfaction of having simplified … ". (shrink)

Our normal discourse is replete with discussion of things which do not exist — the objects of fiction, of illusion and hallucination, of religious worship, of misguided fears and other intentional states. Let us call such discourse empty. How to account for the meaning of empty discourse, and such truth values as its statements have, are perennial and thorny philosophical topics. Many positions are well known; in this book of five chapters Azzouni advocates another. Empty discourse is literally about nothing; (...) nonetheless, one may give an account of how its sentences are both meaningful and have appropriate truth values. In the first three chapters, the ideas are applied to the objects of mathematics, hallucination, and fiction, respectively. Chapter 5 fills out the picture with a discussion of truth-conditional semantics. Chapter 4 broaches the question of how all this is relevant to a broader question: the unity of the sciences.The book is ingenious and clearly argued.1 There are many perceptive discussions — for example, of the way we talk about fictional objects, and about hallucination. And although it is not the main aim of the book, it mounts a number of telling arguments against alternative positions. And even if the book did not persuade at least this reader of its main thesis, it has estabished a genuinely novel and intriguing position …. (shrink)

The general question considered is whether and to what extent there are features of our mathematical knowledge that support a realist attitude towards mathematics. I consider, in particular, reasoning from claims such as that mathematicians believe their reasoning to be part of a process of discovery (and not of mere invention), to the view that mathematical entities exist in some mind-independent way although our minds have epistemic access to them.

Moritz Pasch is usually seen today as a precursor of Hilbert. The Vorlesungen über neuere Geometrie is indeed one of the few works Hilbert referred to in the Grundlagen. Unfortunately, Hilbert's epoch-making book has eclipsed Pasch's achievement; so much so that Pasch's Vorlesungen has not been yet translated into English. But as Pollard emphasizes, it would be a mistake to reduce Pasch's research to his work on the foundations of geometry. Pasch published another book on the foundations of real analysis (...) and arithmetic in 1882, and two other works, respectively, in 1909 and in 1914 . Pasch thus aimed not only to analyze the foundations of geometry but also to elaborate a complete theory of mathematical knowledge. This ambition brought him close to the logicists . But, unlike them, Pasch was a resolute empiricist: he considered that ‘geometry and number theory are branches of empirical science’ 1 and that all mathematical concepts must be derived from a set of notions abstracted from our experience.Pasch made his whole career in Giessen — that is, outside of the major German research centers. Heavily burdened with administrative duties, Professor Pasch seems to have found time for his research only in 1917, when he retired at the age of 67. As Pollard notes, nearly half of Pasch's published output appeared after his sixty-eighth birthday! This volume is a translation of fourteen articles dating from the later period of Pasch's life, ordered by the time of their publication. The papers are preceded by a substantial introduction by Stephen Pollard, the editor and translator of all the papers. Aside from his correspondence with Frege, this volume is thus the first translation of Pasch's work …. (shrink)

According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. When this grand narrative was brought into question, our perspectives on the question of mathematization should have changed. It seems, however, that they were instead set aside, both because of a general distrust towards sweeping narratives that are always subject to the suspicion that they overlook the unyielding complexity of real history, (...) and because of a shift in our interests. The more obscure and idiosyncratic they are, an alchemist, a patron of the sciences or a lunatic collector is nowadays honored in journals of the history of sciences. As for the general issues involved in the question of mathematization, they are rejected as obsolete, or reserved for specialized journals in the history of mathematics. The article « Forms of mathematization » examines in general the notion of mathematization, distinguishes different forms of mathematization and defends a renewed study of the forms of mathematization in the history of the early sciences. (shrink)

(No abstract is available for this citation).

This book provides an accessible, critical introduction to these three projects as it describes and investigates both their philosophical and their mathematical ...

Is alternative mathematics possible? More specifically, is it possible to imagine that mathematics could have developed in any other than the actual direction? The answer defended in this paper is yes, and the proof consists of a direct demonstration. An alternative mathematics that uses vague concepts and predicatesis outlined, leading up to theorems such as "Small numbers have few prime factors''.

This ninth volume in the Library of Exact Philosophy series is a development of the author’s 1971 McGill University dissertation written under the guidance of Mario Bunge. The thesis of the book is that the objectivity of mathematics does not require that there be any mathematical objects. The objectivity of mathematics is the widespread agreement among working mathematicians on what is provable, i.e., on what entailments hold between mathematical constructs. Castonguay gives precise definitions of several terms; but, unfortunately, "construct" is (...) not one of them. He has set himself the task of arguing that we can account for community recognition and acceptance of proofs without assuming that working mathematicians have had to inspect some special mathematical objects to verify that things are in the mathematical realm as they are said to be in the proved propositions. He does concede that many mathematicians, for heuristic reasons, may need to speak as if there were a mind-independent mathematical realm. (shrink)

The main thesis of this paper is that, Contrary to general belief, George berkeley did in fact express a coherent philosophy of mathematics in his major published works. He treated arithmetic and geometry separately and differently, And this paper focuses on his philosophy of arithmetic, Which is shown to be strikingly similar to the 19th and 20th century philosophies of mathematics known as 'formalism' and 'instrumentalism'. A major portion of the paper is devoted to showing how this philosophy of mathematics (...) follows directly from berkeley 's theory of signs and his philosophy of language, And from his discovery of an alternative to the correspondence theory of truth used by most of his predecessors. (shrink)

THE notion of quantity is basic and it is no surprise that Aristotle refers to it in many places. There are two main discussions, that in the Categories—a part of the Organon which is of great interest to modern logicians and that spread over the physical treatises. Naturally the two treatments overlap, but modern logic is at a far remove from classical cosmology and it is fairly easy to separate them at their sources. This I have attempted to do by (...) considering quantity as an attribute and as divisible. There follows a brief consideration of the notions arrived at for their relevance to modern mathematics with particular reference to the real number system and the notion of infinity. (shrink)

The generality relativist has been accused of holding a self-defeating thesis. Kit Fine proposed a modal version of generality relativism that tries to resist this claim. We discuss his proposal and argue that one of its formulations is self-defeating.

(ENG) The paper examines the first attempts put forward in ancient Greek to build a method for scientific discovery and how they have been progressively neglected. -/- (ITA) L'articolo esamina i primi tentativi effettuati nell'antica grecia di costruire un metodo della scoperta scientifica, e analizza le ragioni che hanno portato al suo progessivo abbandono.